Manuscript En05-C@1 · En05-C.tex · section · aa213ea6

Stage H: a constructed bounded-sequence observable limit

Stage H: a constructed bounded-sequence observable limit · lines 1827–1863

En05-C@1#eq:sequencequotient, En05-C@1#eq:quotientnorm, En05-C@1#eq:limitlocalalgebra

This section gives an actual observable algebra, not a conclusion from commuting superoperators or convergent outcome numbers.

Let $M_{n}$ be unital $C^{*}$ -algebras. Define

\prod_{b} M_{n} = {(a_{n}) : n sup ∥ a_{n} ∥ < \infty}, ⨁_{0} M_{n} = {(a_{n}) : ∥ a_{n} ∥ \to 0} .

The latter is a closed two-sided ideal. Put

M_{\infty} = (\prod_{b} M_{n}) / (⨁_{0} M_{n}) .

Write $[a_{n}]$ for a class. Its norm is

∥[a_{n}]∥ = n lim sup ∥ a_{n} ∥ .

For nets $A_{n} (O) \subseteq M_{n}$ , define

A_{\infty} (O) = {[a_{n}] : a_{n} \in A_{n} (O), n sup ∥ a_{n} ∥ < \infty} .

Displayed formulas in this passage

unlabeled · line 1834
$\prod_{b} M_{n} = {(a_{n}) : n sup ∥ a_{n} ∥ < \infty}, ⨁_{0} M_{n} = {(a_{n}) : ∥ a_{n} ∥ \to 0} .$
En05-C@1#eq:sequencequotient · line 1844
$M_{\infty} = (\prod_{b} M_{n}) / (⨁_{0} M_{n}) .$
En05-C@1#eq:quotientnorm · line 1852
$∥[a_{n}]∥ = n lim sup ∥ a_{n} ∥ .$
En05-C@1#eq:limitlocalalgebra · line 1858
$A_{\infty} (O) = {[a_{n}] : a_{n} \in A_{n} (O), n sup ∥ a_{n} ∥ < \infty} .$